Torsion Points on Elliptic Curves over Quartic Fields

William Stein

(this is joint work with Sheldon Kamienny)

University of Washington

May 2010

Motivating Problem

Let $K$ be a number field.  

Theorem (Mordell-Weil): If $E$ is an elliptic curve over $K$, then $E(K)$ is a finitely generated abelian group.

Thus $E(K)_{\rm tor}$ is a finite group. 

Problem:  Which finite abelian groups $E(K)_{\rm tor}$ occur, as we vary over all elliptic curves $E/K$?

Observation: $E(K)_{\rm tor}$ is a finite subgroup of $\CC/\Lambda$, so $E(K)_{\rm tor}$ is cyclic or a product of two cyclic groups.

An Old Conjecture

Conjecture (Levi around 1908; re-made by Ogg in 1960s): 

  When $K=\QQ$, the groups $E(\QQ)_{\rm tor}$, as we vary over all $E/\QQ$, are the following 15 groups:

    $\ZZ/m\ZZ$                            for $m\leq 10$ or $m=12$

    $(\ZZ/2\ZZ) \times (\ZZ/2v\ZZ)$    for $v\leq 4$.

Note:

1. This is really a conjecture about rational points on certain curves of (possibly) higher genus 
2. Or, it's a conjecture in arithmetic dynamics about periodic points.

Modular Curves

The modular curves $Y_0(N)$ and $Y_1(N)$:

• Let $Y_0(N)$ be the affine modular curve over $\QQ$ whose points parameterize isomorphism classes of pairs $(E,C)$, where $C \subset E$ is a cyclic subgroup of order $N$.
• Let $Y_1(N)$ be ...  of pairs $(E,P)$, where $P\in E(\overline{\QQ})$ is a point of order $N$.

Let $X_0(N)$ and $X_1(N)$ be the compactifications of the above affine curves.

Observation: There is an elliptic curve $E/K$ with $p \mid \#E(K)_{\rm tor}$ if and only if $Y_1(p)(K)$ is nonempty.

Also, $Y_0(N)$ is a quotient of $Y_1(N)$, so if $Y_0(N)(K)$ is empty, then so is $Y_0(N)$. 

Mazur's Theorem (1970s)

Theorem (Mazur) If $p \mid \#E(\QQ)_{\rm tor}$ for some elliptic curve $E/\QQ$, then $p\leq 13$.

Combined with previous work of Kubert and Ogg, one sees that Mazur's theorem implies Levi's conjecture, i.e., a complete classification of the finite groups $E(\QQ)_{\rm tor}$.

Here are representative curves by the way (there are infinitely many for each $j$-invariant):

Mazur's Method

Theorem (Mazur) If $p \mid \#E(\QQ)_{\rm tor}$ for some elliptic curve $E/\QQ$, then $p\leq 13$.

Basic idea of the proof:  

1. Find a rank zero quotient $A$ of $J_0(p)$ such that...
2. ... the induced map $f:X_0(p) \to A$ is a formal immersion at infinity (this means that the induced map on complete local rings is surjective). 
3. Then consider the point $x \in Y_0(p)$ corresponding to a pair $(E,\langle P \rangle)$, where $P$ has order $p$.  
4. If $E$ has potentially good reduction at $3$, get contradiction by injecting $p$-torsion mod $3$ since $p>13$, so $E$ has multiplicative reduction.  We may thus assume $x$ reduces to the cusp $\infty$. 
5. The image of $x$ in $A(\QQ)$ is thus in the kernel of the reduction map mod $3$.     But this kernel of reduction is a formal group, hence torsion free.  But $A(\QQ)=A(\QQ)_{\rm tor}$ is finite, so image of $x$ is 0. 
6. Use that $f$ is a formal immersion at infinity along with step 5, to show that $x=\infty$, which is a contradiction since $x\in Y_0(p).$

Mazur uses for $A$ the Eisenstein quotient of $J_0(p)$ because he is able to prove -- way back in the 1970s! -- that this quotient has rank $0$ by doing a $q$-descent, for primes $q$ that divide the numerator of $(p-1)/12$.   This is long before much was known toward the BSD conjecture.  More recently one can:

• Merel 1995: use the bigger winding quotient of $J_0(p)$, which is the maximal analytic rank $0$ quotient.  This makes the arguments easier, and we know by Kolyvagin-Logachev et al. or by Kato that the winding quotient has rank 0.  (For $p=67$ the winding and Eisenstein quotients already differ, since 67a has trivial torsion and rank 0.)
  
  
• Parent 1999: use instead the winding quotient of $J_1(p)$, which leads to a similar argument as above.  This quotient has rank 0 by Kato's theorem.  

Kamienny-Mazur

A prime $p$ is a torsion prime for degree $d$ if there is a number field $K$ of degree $d$ and an elliptic curve $E/K$ such that $p \mid \#E(K)_{\rm tor}$. 

Let $S(d) = \{ \text{torsion primes for degree } \leq d \}$.  

For example, $S(1) = \{2,3,5,7\}$. 

Finding all possible torsion structure over all fields of degree $\leq d$ often involves determining $S(d)$, then doing some additional work (which we won't go into).  E.g.,

Theorem (Frey, Faltings): If $S(d)$ is finite, then the set of groups $E(K)_{\rm tor}$, as $E$ varies over all elliptic curves over all number fields $K$ of degree $\leq d,$ is finite. 

Kamienny and Mazur: Replace $X_0(p)$ by the symmetric power $X_0(p)^{(d)}$ and gave an explicit criterion in terms of independence of Hecke operators (or $q$-expansions of modular forms) for $f_d: X_0(p)^{(d)} \to J_0(p)$ to be a formal immersion at $(\infty, \infty,\ldots,\infty)$.   A point $y\in X_0(p)(K)$, where $K$ has degree $d$, then defines a point $\tilde{y} \in X_0(p)^{(d)}(\QQ)$, etc.

Theorem (Kamienny and Mazur):

• $S(2) = \{2,3,5,7,11,13\}$,
• $S(d)$ is finite for $d\leq 8$,
• $S(d)$ has density 0 for all $d$.

Abromovich soon proved that $S(d)$ is finite for $d\leq 14$. 

Corollary (Uniform Boundedness): There is a fixed constant $B$ such that if $E/K$ is an elliptic curve over a number field of degree $\leq 8$, then $\# E(K)_{\rm tor} \leq B$.

(Very surprising!)

Torsion Structures over Quadratic Fields

Theorem (Kenku, Momose, Kamienny, Mazur): The complete list of subgroups that appear over quadratic fields is:

            Z/mZ              for m <= 16 or m = 18
           (Z/2Z) x (Z/2vZ)   for v <= 6.
           (Z/3Z) x (Z/3vZ)   for v = 1,2
           (Z/4Z) x (Z/4vZ)

and each occurs for infinitely many $j$-invariants.

What is $S(d)$?

Kamienny, Mazur: "We expect that $\max(S(3)) \leq 19$, but it would simply be too embarrassing to parade the actual astronomical finite bound that our proof gives."

But soon, Merel in a tour de force, proves (by using the winding quotient and a deep modular symbols argument about independence of Hecke operators):

Theorem (Merel, 1996):  $\max(S(d)) < d^{3 d^2}$, for $d\geq 2$.

thus proving the full Universal Boundedness Conjecture, which is a huge result.

Shortly thereafter Oesterle modifies Merel's argument to get a much better upper bound:

Theorem (Oesterle): $\max(S(d)) \leq (3^{d/2}+1)^2$.

Remark (Merel, personal communication, 2010-05-10)

1. The known bounds for $S(d)$ are exponential in $d$.  However, a polynomial bound on $S(d)$ in $d$ is expected. Therefore, one can not expect to computationally determine the exact list of torsion primes in degree for many more $d$'s. 
2. The bound is obtained by considering (essentially) two cases (according to the type of reduction modulo $\ell$ of your elliptic curve) : in one case it is easily seen to be exponential in $d$, the hard case finally yields a bound which is polynomial in $d$ (something like $O(d^8)$ in my paper, $O(d^6)$ after Oesterlé, I suspect one can lower it to $O(d^2)$). Unsatisfying!
3. If you want a bound depending on the field $K$ (instead of just the degree of $K$), you can obtain a bound like O(size of the residue field of $K$ of smallest order).

Parent's Kamienny Method: Nailing Down S(3)

By Oesterle, we know that $\max(S(3)) \leq 37$.  

In 1999, Parent made Kamienny's method applied to $J_1(p)$ explicit and computable, and used this to bound $S(3)$ explicitly, showing that $\max(S(3)) \leq 17.$   This makes crucial use of Kato's theorem toward the Birch and Swinnerton-Dyer conjecture!  

In subsequent work, Parent rules out $17$ finally giving the answer:

$$  S(3) = \{2,3,5,7,11,13\}  $$

The list of groups $E(K)_{\rm tor}$ that occur for $K$ cubic is still unknown.  However, using the notion of trigonality of modular curves (having a degree 3 map to $P^1$), [Jeon, Kim, and Schweizer, 2004] showed that the groups that appear for infinitely many $j$-invariants are:

    Z/mZ           for m<=16, 18, 20
    Z/2Z x Z/2vZ   for v<=7

Remark: Parent also gave an explicit bound for the torsion of order powers of prime numbers in his thesis...

What about Degree 4?

By Oesterle, we know that $\max(S(4)) \leq 97$.

Recently, Jeon, Kim, and Park (2006), again used gonality (and big computations with Singular), to show that the groups that appear for infinitely many $j$-invariants for curves over quartic fields are:

    Z/mZ           for m<=18, or m=20, m=21, m=22, m=24
    Z/2Z x Z/2vZ   for v<=9
    Z/3Z x Z/3vZ   for v<=3
    Z/4Z x Z/4vZ   for v<=2
    Z/5Z x Z/5Z 
    Z/6Z x Z/6Z

Question: Is $S(4) = \{2,3,5,7,11,13,17\}?$

Explicit Kamienny-Parent for $d=4$

To attack the above unsolved problem about $S(4)$, I made Parent's (1999) approach very explicit in case $d=4$ and $\ell=2$ (he gives a general criterion for any $d$...).  One arrives that the following (where $t$ is a certain explicitly computable element of the Hecke algebra).    With $\ell=2, d=4$, we have $(1+\ell^{d/2})^2=25$.

NOTES:

1. This looks pretty crazy, but this is really just a way of expressing the condition that a certain map is a formal immersion. 
2. As $p$ gets large, there are a LOT of 4-tuples of elements of the Hecke algebra to test for independence mod 2.
3. Here is code that implements this algorithm: code.sage

Running the Algorithm

After a day we find that the criterion is not satisfied for $p=29,31$, but it is for $37\leq p \leq 97$. 

Conclusion:

Theorem (Kamienny, Stein):  $\max(S(4)) \leq 31$. 

Last Week...

A complete solution!?!

Theorem (Kamienny, Stein, Stoll): $S(4) = \{2,3,5,7,11,13,17\}$

Proofs uses that ${\rm rank}(J_1(p))=0$ for the above $p$, informed by calculations from [Conrad-Edixhoven-Stein] about the arithmetic of $J_1(p)$ for small $p$, so one can use much more direct geometric arguments.    This also involves some large computations with Magma on explicit algebraic curves, e.g., Riemann-Roch spaces, enumerating and reducing divisors, etc., built on top of Florian Hess's function fields package.  Stoll: "Finding the degree 4 points takes about 3 hours [...]  The other problem is that Magma crashes once in a while when turning a point into a place. This will be fixed in the next release, but for now, one may have to try the actual checking a few times until it runs through."  

Related Conjecture (Stein): $J_1(p)(\QQ)_{\rm tor}$ is generated by differences of rational cusps.

(See extensive data about this conjecture in Conrad-Edixhoven-Stein.)

Future Work

1. Determine if $J_1(p)(\QQ)_{\rm tor}$ is cuspidal.
2. Make the algorithm for showing that $\max(S(4)) \leq 31$ much more efficient.  Then...
3. Repeat my calculations, but for $d=5$ and hope to replace the Oesterle bound of $\max(S(5)) \leq 271$ by $$\max(S(5)) \leq 43 \quad\text{  (or something close)}.$$
4. Isogeny degrees -- still an open problem even over quadratic fields!  
   • Cremona (a few minutes ago on Google Buzz): "I'm also very interested in the corresponding question for $X_0(\ell)$, so we know what the possible prime degrees of isogenies are for a given field (or degree). I had some interesting correspondence about this with Parent about 6 months ago; he says that is still wide open for quadratic fields!    My student Kimi is implementing isogenies of degree 11, 17, 19 (the genus 1 cases) in Sage (work in progress). But to have a genuine isogeny_class() function over any non-Q number fields we need a bound."   and
   • Mazur (email): "It would be also interesting if you could, say, rule out a few  primes $p$ occurring as $p$-isogenies over such fields (for non CM curves)?"
5. Modular abelian varieties